is the Frobenius–Schur indicator for \(\chi \) .
It is well known that
\(\nu _2(\chi )\in \lbrace -1,0,1\rbrace \) , and that this value is nonzero if and only if \(\chi \)  is a real-valued character, and it is  1 exactly if \(\chi \)  is the character of a real representation of \(G\)  (see [1]} or [2]}).
Thus, when \(\chi \)  is the character of a real representation of \(G\) ,
we have
\(1= \nu _2(\chi ) = \mbox{$\langle \chi ^{(2)},\mbox{\bf 1}_G  \rangle $} = \mbox{$\langle \chi _S - \chi _A, \mbox{\bf 1}_G \rangle $},\) 
